gen-define-the-t-distribution-and-give-one-short-worked-example-ddfe

Define the t-distribution and give one short worked example.

# The t-Distribution ## Definition The **t-distribution** (or Student's t-distribution) is a probability distribution used for hypothesis testing and confidence intervals when: - The population standard deviation is **unknown** - The sample size is **small** (typically n < 30) - The population is approximately normally distributed It has one parameter: **degrees of freedom (df)** = n - 1 **Key properties:** - Bell-shaped and symmetric (like normal distribution) - Heavier tails than the normal distribution - Approaches the normal distribution as df increases --- ## Worked Example **Problem:** A coffee shop claims their espresso machine dispenses 30 mL per shot. You measure 6 shots and get: 28, 31, 29, 30, 32, 28 mL. Test if the true mean differs from 30 mL (α = 0.05). **Solution:** 1. **Calculate sample statistics:** - Sample mean: x̄ = (28+31+29+30+32+28)/6 = 29.67 mL - Sample std dev: s = 1.63 mL - n = 6, so df = 5 2. **Test statistic:** t = (x̄ - μ₀)/(s/√n) = (29.67 - 30)/(1.63/√6) = -0.33/0.67 = **-0.49** 3. **Critical value:** - Two-tailed test, α = 0.05, df = 5 - t₀.₀₂₅,₅ = ±2.571 4. **Conclusion:** Since |-0.49| < 2.571, we **fail to reject** H₀. No significant evidence the mean differs from 30 mL.